Classification of the linearly reductive finite subgroup schemes of $S{L}_2$

We classify the linearly reductive finite subgroup schemes $G$ of $S{L}_2=SL(V)$ over an algebraically closed field $k$ of positive characteristic, up to conjugation. As a corollary, we prove that such $G$ is in one-to-one correspondence with an isomorphism class of two-dimensional $F$-rational Gorenstein complete local rings with the coefficient field $k$ by the correspondence $G\mapsto ((V{)}^G)\widehat{}$.